58 A^VÇÚO 1n ò f S can be represented as S = Z, 1.1) U 1/ where Z N, 1) and U U, 1) are ndependent, > s the shape parameter. The dstrbuton was named b

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1 A^VÇÚO 1n ò 1 Ï 15c Chnese Journal of Appled Probablt and Statstcs Vol.31 No.1 Feb. 15 An Exponental Slash Half Normal Dstrbuton for Analzng Nonnegatve Data Xu Mepng Mathematcs Department, School of Scence, Bejng Technolog and Busness Unverst, Bejng, 148) Gu Wenhao Department of Mathematcs and Statstcs, Unverst of Mnnesota Duluth, Mnnesota State, USA, 5581) Abstract A new class of slash dstrbuton s studed for analzng nonnegatve data. Ths dstrbuton s defned b means of a stochastc representaton as the mxture of a half normal random varable wth the power of an exponental random varable. Denst functon and propertes nvolvng hazard functon, moments and moment generatng functon are derved. The usefulness and flexblt of the proposed dstrbuton are llustrated through a real applcaton b maxmum lkelhood procedure. Kewords: Exponental slash half normal dstrbuton, slash half normal dstrbuton, kurtoss coeffcent, skewness coeffcent. AMS Subject Classfcaton: 6E15, 6F Introducton The normal dstrbuton plas a ke role n statstcal analss. However, the nference based on the normal dstrbuton s senstve to statstcal errors that have a heaver taled dstrbuton Arslan and Genç, 9). Statstcans have actvel been constructng more flexble dstrbutons that can be alternatves to the normal dstrbuton for modelng of data sets nvolvng errors wth heaver tals. One such extenson s slash dstrbuton, whch has been ver popular n robust statstcal analss Rogers and Tuke, 197; Morgenthaler, 1986; Jamshdan, 1; Kashd and Kulkarn, 3). The slash dstrbuton s defned as the rato of two ndependent random varables, whch s descrbed as follows: A random varable S has a standard slash dstrbuton SL) The project was supported b Natonal Natural Scence Foundaton of Chna ) and Bejng Muncpal Part Commttee Organzaton Department talents project 1D535). Receved Jul 1, 14. Revsed September 9, 14. do: /j.ssn

2 58 A^VÇÚO 1n ò f S can be represented as S = Z, 1.1) U 1/ where Z N, 1) and U U, 1) are ndependent, > s the shape parameter. The dstrbuton was named b Rogers and Tuke n ther paper publshed n 197. The SL) denst functon s gven b fx) = 1 u φxu)du, < x <, 1.) where φt) = 1/ π) exp t /} s the standard normal denst functon. The standard slash denst has heaver tals than those of the normal dstrbuton and has large kurtoss. Rogers and Tuke 197) and Mosteller and Tuke 1977) studed the general propertes of the faml. Later, Kafadar 198) dscussed the maxmum lkelhood estmators MLEs) for the locaton-scale case. In recent ears, Wang and Genton 6) developed a multvarate verson and also an asmmetrc multvarate verson and studed ts propertes and nference. Arslan 8) proposed an alternatve asmmetrc extenson of ths dstrbuton b usng the varance-mean mxture of the multvarate normal dstrbuton. Arslan and Genç 9) dscussed a smmetrc extenson of the multvarate verson and Genç 7) dscussed a smmetrc unvarate generalzaton of the slash dstrbuton. Arslan 9) also studed the MLE for the parameters of the skew slash dstrbuton ntroduced n Arslan 8). Gómez et al. 7) defned some smmetrc extensons of the dstrbuton based on ellptcal dstrbutons and studed ts general propertes of the resultng famles, ncludng ther moments. For nonnegatve data, Pewse, 4) studed the asmptotc nference and MLEs for the general locaton-scale half normal dstrbuton, and Wper et al. 8) proposed the Baesan approach for the general half normal and half-t dstrbutons. Olmos et al. 1) defned an extenson based on the half normal dstrbuton and derved some propertes. The clamed a random varable S had a slash half normal dstrbuton SHN, ) f S can be represented as S = Z, 1.3) U 1/ where Z HN) wth denst f Z z; ) = /)φz/), z >. Z s ndependent of U U, 1) and >, >. Recentl, Gu 13) defned another extenson of the half normal dstrbuton, alpha half normal dstrbuton whch s based on a mxture dstrbuton proposed b Elal-Olvero 1), and studed ts propertes. In ths paper, we consder another extended half normal random varable Y and stud ts propertes. As ndcated b computng skewness and kurtoss coeffcents of the new dstrbuton, we are able to show that ths dstrbuton s much more flexble than HN,

3 1 Ï M±? Í: Œ^u ÛšKêâ êš~œ Ù 59 SHN and Gamma dstrbutons n such aspects, and hence s more flexble and approprate for modelng of nonnegatve data sets that ma have heav tals or wth outlers, such as the data from a lfe test that s truncated at a preassgned tme n survval analss or loss data on an asset n rsk management. Other applcaton such as modelng postve errors or usng n regresson dagnose just as HN dstrbuton s also consdered. The rest of ths paper s organzed as follows: In Secton, we ntroduce the new dstrbuton and stud ts propertes, ncludng the stochastc representaton, denst functon, falure rate hazard) functon, moments and moment generatng functon. Secton 3 presents the nference, moments and maxmum lkelhood estmaton for parameters. In Secton 4, a real data set s analzed and reported and ths llustrates that the proposed model fts the real data set well. And we conclude our work n Secton 5.. Exponental Slash Half Normal Dstrbuton and Its Propertes We now gve the general stochastc representaton and the denst functon of the exponental slash half normal dstrbuton, and stud some of ts mportant propertes such as hazard functon, moment generatng functon and moments. For llustratve reason, we also provde some plots of the denst functon, hazard functon, skewness and kurtoss coeffcents for some values of the parameters..1 Stochastc Representaton and Denst Functon A random varable Y s sad to have an exponental slash half normal dstrbuton ESHN, λ, ) f t can be represented as the rato Y = Z,.1) U 1/ where Z HN), U Expλ) wth denst f U x; λ) = λe λx, x >, whch s ndependent of Z and >, λ > and >. b or From.1), we get the denst functon of ESHN, λ, ) as follows. Proposton.1 Let Y ESHN, λ, ). Then the denst functon of Y s gven f Y ;, λ, ) = f Y ;, λ, ) = π λ λe λu φ u 1/ ) u 1/ du.) t exp t + λt )} dt..3)

4 6 A^VÇÚO 1n ò For = 1,.3) can be wrtten as λ λ f Y ;, λ, 1) = exp π } π λ ), f ; 1, f =. λ And for =,.3) becomes a smple formula lke Proof f Y ;, λ, ) = λ + λ ) 3/. In fact, from.1), and usng the ndependence of Z and U, t s eas to see that the denst functon of Y can be wrtten as f Y ;, λ, ) = f Z u 1/ )f U u)u 1/ du. Hence the formula.) follows after replacng f Z ) and f U ) wth ther correspondng denst functons mentoned above. Let t = u 1/,.) s transformed nto.3). For = 1, f =, we have f Y ;, λ, 1) = And f, f Y ;, λ, 1) = π λ = π λ π λ exp λ te λt dt = exp t) } te λt dt } π 1 λ. t exp t + λ /) } dt. Makng the varable transformaton t = / )u λ/ ), hence f Y ;, λ, 1) = π λ exp λ } ue u / λ e u / ) du. Then the result follows b the ntegrals ue u / du = 1 and e u / du = π/. And for =, f Y ;, λ, ) = π λ t ) exp + λ t } dt. Makng the varable transformaton t = u/ + λ ), then f Y ;, λ, ) = π λ ) 3/ + λ u 1/ e u du.

5 1 Ï M±? Í: Œ^u ÛšKêâ êš~œ Ù 61 And the result follows mmedatel b the defnton and propertes of Gamma functon. Consderng the frst and second dervatve of the denst functon, we conclude that the denst functon s decreasng and concave frst and then convex. For example, t has an nflecton pont at = λ/ f =. Fgure 1 shows several plots of the denst functon of ESHN dstrbuton wth dfferent parameters. f,, λ, ) Fgure 1 Remark 1 =. Also, =1,λ =, =.5 =1,λ =, =1 =1,λ =, =5 =1,λ =, f,, λ, ) =1,λ =1, =5 =1,λ =3, =5 =1,λ =5, =5 =1,λ =7, =5 The denst functon of ESHN, λ, ) wth some dfferent parameters From formula.), we can see that f Y ;, λ, ) s contnuous at lm f Y ;, λ, ) = f Z ; ). Whch means that ESHN dstrbuton contans HN dstrbuton as a lmt case when.. Some Propertes Followng b drect computaton, the cumulatve dstrbuton functon CDF) of Y ESHN, λ, ) s gven b F Y ;, λ, ) = = = su 1/ f Y s;, λ, )ds = φ su λe λu 1/ ) u 1/ φ dsdu u λe λu 1/ Φ ) λe λu u 1/ duds ) du 1,.4)

6 6 A^VÇÚO 1n ò for >. Where Φ ) s the CDF of standard normal dstrbuton. The correspondng hazard functon s h Y ;, λ, ) = f Y ;, λ, ) 1 F Y ;, λ, ) = 1 λe λu φ u 1/ ) u 1/ du..5) λe λu [1 Φu 1/ /)]du Fgure shows several plots of the hazard functon of ESHN dstrbuton wth dfferent parameters. We fnd the hazard functon s ncreasng frst and then decreasng slowl, exhbts unmodal upsde down bathtub) shape. h,, λ, ) Fgure =1,λ =, =.5 =1,λ =, =1 =1,λ =, =3 =1,λ =, =5 h,, λ, ) =1,λ =1, =5 =1,λ =3, =5 =1,λ =5, =5 =1,λ =7, =5 The hazard functon of ESHN, λ, ) wth some dfferent parameters Proposton. Let Y U = u HNu 1/ ) and U Expλ). Then Y ESHN, λ, ). Proof In fact, the margnal dstrbuton of Y can be wrtten as f Y ;, λ, ) = So the result follows. f Y U u)f U u)du = ) u 1/ φ λe λu du. Remark Proposton. ndcates that ESHN dstrbuton can be represented as a mxture of a partcular scale HN dstrbuton and the exponental dstrbuton. Ths s an mportant result n the sense that t provdes a smple wa for generatng random numbers from ESHN dstrbuton f ou notce the CDF gven b.4) s an mproper ntegral.

7 1 Ï M±? Í: Œ^u ÛšKêâ êš~œ Ù 63 Proposton.3 MGF) of Y can be represented as Let Y ESHN, λ, ), then the moment generatng functon M Y t) = λ v 1 exp λv + t) )} v dv..6) Proof Usng Proposton., we can wrte the MGF of Y as [ M Y t) = E[Ee ty U)] = E = t) }] exp U / t) } exp u / λe λu du. On the second eual, we use the result that the MGF of HN) s expt) /}. Then makng the varable transformaton v = u 1/ and.6) follows..3 Moments In ths secton, the moments are derved, whch can be used n dervng moments estmators, skewness and kurtoss coeffcents evaluaton. Proposton.4 Let Y ESHN, λ, ), then for r = 1,,... and r <, the rth moment of random varable Y s gven b µ r = EY r r r + 1 ) ) = π λr/ r Γ Γ 1 r ),.7) here Γα) = x α 1 e x dx. Proof Applng.3), we get µ r = π λ r exp t) } t e λt dtd. Changng the order of ntegrals, then µ r = π λ t e λt r exp t) } ddt r + 1 ) = π λ r 1)/ r Γ t r 1 e λt dt r r + 1 ) = π λr/ r Γ Γ 1 r ). The second and thrd euals are b makng the varable transformaton = u/t and t = v/λ) 1/ respectvel.

8 64 A^VÇÚO 1n ò Corollar.1 Let Y ESHN, λ, ), then t follows that EY ) = π λ1/ Γ 1 1 ), > 1; Var Y ) = λ /[ Γ 1 ) π Γ 1 1 ) ], >. skewness β1 Corollar. Let Y ESHN, λ, ), hence the skewness and kurtoss coeffcents of Y are gven b β 1 = πγ 1 3 ) 3πΓ 1 ) Γ 1 1 ) + 4Γ 1 1 ) 3 β = 3π Γ 1 4 ) 16πΓ Remark 3 [ πγ 1 ) Γ 1 1 ) ] 3/, > 3;.8) 1 3 ) Γ 1 1 ) + 1πΓ 1 ) Γ 1 1 ) 1Γ 1 1 ) 4 [ πγ 1 ) Γ 1 1 ) ], > 4..9) From.8) and.9), we notce that both the skewness and kurtoss coeffcents of ESHN dstrbuton are not relatve to scale parameter and λ. And from Fgure 3, we see that both of them are lager than correspondng ones of SHN whch can be found n Olmos et al. 1)) and decrease wth the onl parameter to correspondng ones of HN whch respectvel are 4 π)/π ) 3/ and 3π 4π 1)/π ) ) as expected. These ndcate that ESHN dstrbuton has heaver rght tal than ts HN parent and SHN dstrbuton, and the heaver tal can be obtaned b takng smaller ESHN, λ, ) SHN, ) HN) kurtoss β ESHN, λ, ) SHN, ) HN) Fgure 3 Skewness and kurtoss coeffcents of ESHN, λ, ), SHN, ) and HN)

9 1 Ï M±? Í: Œ^u ÛšKêâ êš~œ Ù Inference In ths secton, we dscuss moment and maxmum lkelhood estmaton for parameters n ESHN dstrbuton. Notcng that that λ 1/ s a scale combnaton, so we let = 1 and onl consder moment estmaton for λ and. Proposton 3.1 Let Y 1,..., Y n be a random sample from the ESHN1, λ, ) dstrbuton, then the moment estmates of λ and for > are gven b π λ = Y ), Γ1 1/ ) here s the soluton of euaton Y /Y = πγ1 /))/Γ1 1/) ), and Y, Y respectvel are the frst and second sample moment. Proof In fact, t follows from.7), EY ) = π λ1/ )Γ 1 1 ) ; EY ) = λ 1/ ) Γ 1 ). 3.1) Hence replace EY r ) b ts correspondng sample moment Y r, r = 1, n 3.1). The results can be obtaned mmedatel b solvng above sstem of euatons. About the maxmum lkelhood estmaton for parameter θ =, λ, ) n ESHN dstrbuton, we have followng regular procedure. Suppose 1,..., n be a group of sample observatons from the ESHN, λ, ) dstrbuton. From.3), we can wrte the log-lkelhood functon as l, λ, ) n logλ) + n log) n log) + n log t t + λt )} ) dt. =1 Hence we get followng maxmum lkelhood estmatng euatons l = n + 1 t n + 3 t + λt )} dt =1 t t + λt )} =, 3.) dt l λ = n λ n t t + λt )} dt =1 t t + λt )} =, 3.3) dt l = n + n t 1 λt ) logt) t + λt )} dt =1 t t + λt )} =. 3.4) dt

10 66 A^VÇÚO 1n ò The observed Fsher nformaton matrx, the negatve of the second partal dervatves of the log-lkelhood, s used to compute the standard errors of the estmates. The matrx s computed n the form where l = n 3 4 n =1 l l λ l Jθ) = l λ l λλ l λ, l l λ l t + t t + λt )} dt t + λt )} dt + 1 t n +4 exp 6 4 t +λt )} dt t + exp =1 t exp t +λt )} t +λt )} dt ) } dt t exp t +λt )}, dt l λ = l λ = 1 t n + 3 t + λt )} dt =1 t t + λt )} dt t + t } l = l = 1 3 n =1 t + exp l λλ = n λ + n t 3 exp =1 t + λt )} dt [ t t + λt )} dt t + 1 λt ) logt) exp t +λt )} dt t exp [ t exp t ] t + λt )} dt t + λt )} dt t + λt )} dt t 1 λt ) logt) exp t + λt )} dt t +λt )} dt t exp t exp t +λt )} dt l λ = l λ = n t 1 λt ) logt) exp =1 t +λt )} dt }, ] t + λt )} dt t t + λt )} dt t +λt )} dt ) }, t +λt )} dt,

11 1 Ï M±? Í: Œ^u ÛšKêâ êš~œ Ù 67 t exp t +λt )} dt t 1 λt ) logt) exp t +λt )} dt } [ t t + λt )} ], dt l = n + n t [1 λt ) logt)] t + λt )} dt =1 t t + λt )} dt t 1 λt ) logt) t + λt )} dt ) } t t + λt )}. dt The maxmum lkelhood estmatng euatons above are not n a smple form. In general, there are no mplct expressons for the estmates. The estmates can be obtaned through some numercal teraton such as Newton-Raphson procedure. In the case where the model contans a locaton parameter, snce data are nonnegatve, ts MLE s the sample mnmum observaton, see Olmos et al. 1). 4. An Illustratve Data Analss The data set s obtaned from RESSET Fnancal Research Database resset.cn/). We collect a total 133 observatons about the postve monthl return unt %) reported b some bank of Chna from the perod of Februar 1991 to September 13. Table 1 Summar for the postve monthl return data sample sze mean standard devaton b 1 b Table 1 summarzes the data set where b 1 s sample skewness coeffcent and b s sample kurtoss coeffcent. Then the data set s ft wth the HN), SHN, ), ESHN, λ, ) and Gammaα, ) dstrbutons and the performance of these dstrbutons s examned. The denst functon of Gammaα, ) s gven b 1 fx) = α Γα) xα 1 exp x }, x >, α >, >. The fttng results are reported n Table. The Akake nformaton crteron AIC) s used to measure the goodness of ft of the models. AIC = k log L, where k s the number of parameters n the model and L s the maxmzed value of the lkelhood functon for the estmated model. From Table, for ths data, the ESHN model has

12 68 A^VÇÚO 1n ò the smallest AIC and hghest lkelhood values, whch means ESHN model has better performance than the other three models. Fgure 4 s the hstogram of the data set, ncludng estmated denstes under the dstrbutons mentoned above. Fgure 4 further shows that ESHN model can effectvel capture the features of the data. Table MLEs wth SD)) of the HN, SHN, ESHN and Gamma models for the postve monthl return data Model λ loglk AIC HN ) SHN ).6333) ESHN ).).114) Model α loglk AIC Gamma Denst.86).4) 4 6 Hstogram of x ESHN SHN HN Gamma x Fgure 4 Hstogram and ftted curves for the postve monthl return dataset

13 1 Ï M±? Í: Œ^u ÛšKêâ êš~œ Ù Concludng Remarks In ths paper, we have proposed an exponental slash half normal model for nonnegatve data. It s defned to be the rato of two ndependent random varables, a half normal one and a power of the exponental dstrbuton. Ths model contans the half normal model as ts specal case. Propertes nvolvng hazard functon, moments and moment generatng functon are studed. B calculatng the skewness and kurtoss coeffcents we have llustrated the fact that t s able to accommodate nonnegatve data wth hgher skewness and kurtoss. We appl the dstrbuton to the postve monthl return reported b some bank of Chna. Model fttng s mplemented b maxmum lkelhood procedure. The data analss has shown that the proposed model s ver useful n real applcatons and can present a better ft than the half normal and slash half normal, and Gamma models. Fnall, the faml of exponental slash half normal dstrbutons can be further extended b substtutng other random varables from the exponental faml of dstrbutons for the exponental random varable U n.1). Such extenson would provde even more modelng flexblt. For nstance, obtan heaver tals b carefull choosng the varable. But we all know that the models wth more parameters wll become more complext and be not convenent n practcal applcaton. It s not clear though that n dong so we effectvel get extended ranges of kurtoss, our basc motvaton when proposng the exponental slash half normal faml. Such consderatons are currentl subject to addtonal stud. In addton, we can also consder a generaton of our model to a multvarate settng for the further research. References [1] Arslan, O. and Genç, A.İ., A generalzaton of the multvarate slash dstrbuton, Journal of Statstcal Plannng and Inference, 1393)9), [] Rogers, W.H. and Tuke, J.W., Understandng some long-taled smmetrcal dstrbutons, Statstca Neerlandca, 63)197), [3] Morgenthaler, S., Robust confdence ntervals for a locaton parameter: the confgural approach, Journal of the Amercan Statstcal Assocaton, 81394)1986), [4] Jamshdan, M., A Note on parameter and standard error estmaton n adaptve robust regresson, Journal of Statstcal Computaton and Smulaton, 711)1), [5] Kashd, D.N. and Kulkarn, S.R., Subset selecton n multple lnear regresson wth heav taled error dstrbuton, Journal of Statstcal Computaton and Smulaton, 7311)3), [6] Mosteller, F. and Tuke, J.W., Data Analss and Regresson: A Second Course n Statstcs, Addson- Wesle, New Jerse, 1977.

14 7 A^VÇÚO 1n ò [7] Kafadar, K., A bweght approach to the one-sample problem, Journal of the Amercan Statstcal Assocaton, 77378)198), [8] Wang, J. and Genton, M.G., The multvarate skew-slash dstrbuton, Journal of Statstcal Plannng and Inference, 1361)6), 9. [9] Arslan, O., An alternatve multvarate skew-slash dstrbuton, Statstcs and Probablt Letters, 7816)8), [1] Genç, A.İ., A generalzaton of the unvarate slash b a scale-mxtured exponental power dstrbuton, Communcatons n Statstcs - Smulaton and Computaton, 365)7), [11] Arslan, O., Maxmum lkelhood parameter estmaton for the multvarate skew-slash dstrbuton, Statstcs and Probablt Letters, 79)9), [1] Gómez, H.W., Quntana, F.A. and Torres, F.J., A new faml of slash-dstrbutons wth ellptcal contours, Statstcs and Probablt Letters, 777)7), [13] Pewse, A., Large-sample nference for the general half-normal dstrbuton, Communcatons n S- tatstcs - Theor and Methods, 317)), [14] Pewse, A., Improved lkelhood based nference for the general half-normal dstrbuton, Communcatons n Statstcs - Theor and Methods, 33)4), [15] Wper, M.P., Grón, F.J. and Pewse, A., Objectve Baesan nference for the half-normal and half-t dstrbutons, Communcatons n Statstcs - Theor and Methods, 37)8), [16] Olmos, N.M., Varela, H., Gómez, H.W. and Bolfarne, H., An extenson of the half-normal dstrbuton, Statstcal Papers, 534)1), [17] Gu, W.H., An alpha half normal slash dstrbuton for analzng nonnegatve data, Communcatons n Statstcs - Theor and Methods, Accepted, 13. [18] Elal-Olvero, D., Alpha-skew-normal dstrbuton, Proeccones Journal of Mathematcs, 93)1), 4 4. Œ^u ÛšKêâ M± óûœænæêæx,, 148)? Í êš~œ Ù ²Z ˆŒÆÚ d êæúox, ²Z ˆ², I, 5581) ïä «Œ^u ÛšKêâ# Ùx. ^ ÅL«5½Â, Œ ÅCþ ê ÅCþ Ü. ٠ݼêÚ5Ÿ, )ºx¼ê, ÝÚÝ1¼êÑ. T Ùx ^ 5Ú¹5ÏL ~^ ' c: Æ a Ò: Œ, S². êš~œ Ù, š~œ Ù, ÝXê, ÝXê. O1.1.